On a jump Markovian model for a gene regulatory network




De La Chevrotière, Michèle

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We present a model of coupled transcriptional-translational ultradian oscillators (TTOs) as a possible mechanism for the circadian rhythm observed at the cellular level. It includes nonstationary Poisson interactions between the transcriptional proteins and their affined gene sites. The associated reaction-rate equations are nonlinear ordinary differential equations of stochastic switching type. We compute the deterministic limit of this system, or the limit as the number of gene-proteins interactions per unit of time becomes large. In this limit, the random variables of the model are simply replaced by their limiting expected value. We derive the Kolmogorov equations — a set of partial differential equations —, and we obtain the associated moment equations for a simple instance of the model. In the stationary case, the Kolmogorov equations are linear and the moment equations are a closed set of equations. In the nonstationary case, the Kolmogorov equations are nonlinear and the moment equations are an open-ended set of equations. In both cases, the deterministic limit of the moment equations is in agreement with the deterministic state equations.



Genetic Oscillator, Circadian Rhythm, Markov Process, Stochastic Switching System, Kolmogorov Equations, Moment Equations